Complexity of the Single Vehicle Scheduling Problem on Graphs

Let G = (V;E) be a graph. The travel times w(u; v) and w(v; u) are associated with each edge fu; vg 2 E, and a job, which is also denoted as v, is located at each node v 2 V . Each job v has release time r(v) and handling time h(v). A start node s 2 V and a set Q V of goal nodes are speci ed. The VSP (Vehicle Scheduling Problem) asks to nd a routing schedule of the single vehicle that starts from node s at time 0 and reaches one of the goal nodes in Q after visiting all jobs v 2 V for processing. The processing of a job v cannot be started before its release time t = r(v) (hence the vehicle has to wait if it arrives at v before r(v) for processing) and takes h(v) time units once its processing has started (no interruption is allowed). The objective is to nd a schedule that minimizes the completion time (i.e., the time to process all jobs). We rst establish a dynamic programming algorithm for solving VSP on a general graph, and show that VSP can be solved in O(n b ) time when G is a tree with b leaves and all handling times are zero. We then prove that VSP is strongly NP-hard even if graph G is a general tree and all handling times are zero. This research was partially supported by the Scienti c Grant-in-Aid from Ministry of Education, Science and Culture of Japan, and the subsidy from the Inamori Foundation.